Communication and Cooperation in Markets

S. Nageeb Ali and David A. Miller*

February 2, 2021

Abstract

Many markets rely on traders truthfully communicating who has cheated in the pastand ostracizing those traders from future trade. This paper investigates when truthfulcommunication is incentive compatible. We find that if each side has a myopic incen-tive to deviate, then communication incentives are satisfied only when the volume oftrade is low. By contrast, if only one side has a myopic incentive to deviate, then com-munication incentives do not constrain the volume of supportable trade. Accordingly,there are strong gains from structuring trade so that one side either moves first or hasits cooperation guaranteed by external enforcement.

*Ali: Pennsylvania State University. Miller: University of Michigan. We thank Ben Golub and Yu-val Heller for their useful comments. We benefited from constructive feedback from the Editor and threeanonymous referees. This research was financially supported by NSF grant SES–1127643.

1 IntroductionIn many markets, buyers and sellers can renege on their promises without suffering legalconsequences, but defectors are punished by the loss of future business. If a seller tradeswith many buyers, losing business with a single buyer may not be enough of a threat todeter her from deviating. But if cheating a single buyer results in her losing business withmany buyers, then she is more inclined to cooperate. Such schemes, where actions witha single player affect cooperation with others, are at the core of multilateral enforcementor “third party” punishment. Multilateral enforcement schemes often employ personalizedpunishment, where traders will work with those who are untainted but sever their ties tothose who have deviated in the past.

For personalized punishment to work, traders need to be able to communicate with eachother about their past experiences. Scholars have noted how information-sharing institutionswere critical to medieval trade and trust (Milgrom, North and Weingast 1990; Greif 2006).Today, online markets rely on ratings and reviews to collect and disseminate informationabout the behavior of market participants. Credit markets through the ages have benefitedfrom sharing information about borrower histories.

We view communication not as a mechanical process, but as a voluntary choice. Iftraders are unwilling to speak truthfully about their past experiences, information will notflow from one relationship to another, making personalized punishment impossible. Thus,we ask: when do buyers and sellers have a motive to tell the truth to other traders?

We pose this question in a networked market of buyers and sellers, wherein each buyer-seller pair has a long-term trading relationship. Interactions within each relationship arenot directly observed by third parties. When a pair interacts, the seller chooses how muchquality (or quantity) to deliver to the buyer and the buyer chooses how much to pay theseller. The seller faces an increasing cost function, and thus may have an incentive to shirk;the buyer analogously may have an incentive to shortchange the seller. In addition to theseeconomic interactions, sellers also randomly meet other sellers, and buyers randomly meetother buyers, just to share information about their past experiences.

We emphasize that parties “may” have an incentive to deviate because whether a partyactually has an incentive to do so depends subtly on the timing of trade. If the buyer andseller act simultaneously, then each party has a myopic incentive to shirk. But if the rules ofthe marketplace direct the buyer to first make a payment, and the seller then to choose thequantity to trade, then the buyer gains nothing by paying less than the proposed amount,since the seller could then withhold the product. Only the seller has an incentive to shirk.

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Or the timing might be reversed so that the buyer submits payment only after receipt of theproduct, in which case only he has an incentive to deviate. Thus, the trading interactionmay feature two-sided or one-sided moral hazard, depending on the timing of trade.

One reason this distinction between one-sided and two-sided moral hazard matters isthat one-sided moral hazard relaxes the incentive constraints on one side of the market, andthereby offers the potential for increasing the level of cooperation. We find that this differ-ence is amplified by multilateral enforcement for a new reason: it permits some players toshare information about others without having to worry about its consequences. We expositthis logic using the class of permanent ostracism equilibria, and study how these equilibriaperform at a fixed discount rate. We find that permanent ostracism supports substantiallymore cooperation with one-sided moral hazard than with two-sided moral hazard.

What is permanent ostracism? It embodies the idea that a trader, Ann, ceases to tradewith another, Bob, if she comes to learn that Bob has cheated in the past; however, Ann con-tinues trading with all partners whose reputations are untainted from her perspective. Westudy these equilibria for two reasons. First, its description matches market behavior wherepunishments are targeted towards a defector without making the entire market unravel.1

Second, permanent ostracism offers the simplest scheme in which traders’ reputations andrecords are used to punish or reward them. It is a personalized form of grim trigger punish-ment, where if a trader cheats another, it causes a breakdown of cooperation among thosewho come to learn about that deviation. Thus, it has been the focus of many prior papers,most of which abstract from communication incentives. Our focus is on the effectiveness ofthese equilibria when traders strategically communicate about who is guilty and innocent.

We study permanent ostracism equilibrium at a fixed discount rate, and compare it totwo benchmarks. The first benchmark is the lower bound of bilateral enforcement, whichis the most a buyer-seller pair could credibly trade without any third-party punishment.The second benchmark is the upper bound of naive communication, which is the highestlevel of trade achievable if all non-defectors were forced to tell the whole truth, regardless ofincentives. Our main message is the following:

If each buyer-seller pair faces one-sided moral hazard, then permanent ostracismcan achieve the benchmark of naive communication. In contrast, if each buyer-seller pair faces two-sided moral hazard, no permanent ostracism equilibriumsupports more trade than bilateral enforcement.

1This targeting of punishments towards defectors distinguishes permanent ostracism from contagion(Kandori 1992), where innocent players shirk on all others once cheated.

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This result has a clear strategic intuition. In multilateral enforcement, each trader takeson the role of “guarding” each other by letting other market participants know if they observeany defections. However, a trader can be trusted only to the extent that he has more tolose in the future than he can gain by defecting. This raises the classical question of “whowill guard the guardians?” Can market participants be trusted to truthfully reveal whenothers are guilty or must they themselves be guarded?2 With two-sided moral hazard, eachside guards the other, so a trader is unwilling to reveal that she has been cheated becauseit reduces the degree to which she herself can be trusted. With one-sided moral hazard, incontrast, first-movers have no myopic incentive to shirk. This feature permits us to constructequilibria in which the continuation play of that first-mover is decoupled from the messagethat player sends. Hence, these players are willing to communicate truthfully. Effectively,these players become guards who themselves need not be guarded, and their guardianshipsecures the cooperation of others. This difference is sufficiently stark that our negative resultholds even when traders obtain verifiable evidence and our positive result obtains even whentraders’ communication is cheap talk.

The purpose of our paper is to exposit this result using a simple model that highlightsthe role of communication incentives. We describe the model in Section 2. Section 3 brieflydiscusses the negative result that obtains for two-sided moral hazard, primarily as a bench-mark for what follows. Section 4 poses and proves our main result that highlights how theshift to one-sided moral hazard (as obtains in a sequential extensive-form) allows the marketto support greater cooperation. A broader implication of this result is that if word-of-mouthcommunication is to play a role in supporting trade, there are significant gains from struc-turing trade so that one side of the market lacks an incentive to deviate. Doing so amplifiesthe level of supportable trade, because that side of the market can be trusted to truthfullyspread news and information. In Section 5, we illustrate how this framing may be useful tounderstand numerous case studies of when informal enforcement succeeds and fails. We alsodiscuss several important caveats to our analysis.

Related Literature: The role of word-of-mouth communication in trading relationshipshas been studied broadly. Many of these studies document the importance of communication(Greif 1993), or highlight how its speed and dynamics influence cooperation (Raub andWeesie 1990; Klein 1992; Ahn and Suominen 2001; Dixit 2003), but abstract from whethertraders have the motive to communicate truthfully. Analogously, a vast literature discusses

2Hurwicz (2008) poses this question in the context of institutional design, with a focus (like us) onwhether the participants of an institution have incentives to communicate truthfully.

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the role of ratings and feedback in peer-to-peer and online markets (see Tadelis 2016 for asurvey), but in which the willingness of market participants to disclose their past experiencesis often assumed rather than derived.

Our work builds on the study of community enforcement, pioneered by Kandori (1992),Ellison (1994), Harrington (1995), and Okuno-Fujiwara and Postlewaite (1995). One strandof this work envisions that players have “reputational labels”, or records, that are updatedbased on their actions, implicitly assuming that players are sharing information about theirpast experiences.3 This strand of the literature often treats reputational labels as exogenous,and investigates how much information needs to be contained within these labels to facilitatecooperation; see Rosenthal (1979), Takahashi (2010), Heller and Mohlin (2018), Bhaskar andThomas (2019), and Clark, Fudenberg and Wolitzky (2020) for examples.4 Our work studiesthe complementary question of how these labels or records emerge in the first place. Ourresults offer a foundation for truthful records when the interactions involve one-sided moralhazard, and indicates a challenge when the moral hazard is two-sided.

Other recent work studies strategic communication in multilateral enforcement (e.g. Lip-pert and Spagnolo 2011; Bowen, Kreps and Skrzypacz 2013; Wolitzky 2015), illuminatingdifferent facets of the problem but without focusing on this distinction between two-sided andone-sided moral hazard for networked markets. A separate set of papers (Ben-Porath andKahneman 1996; Renault and Tomala 1998; Laclau 2012, 2014) derive general folk theoremswith public or private communication in repeated games with local monitoring structures,whereas our focus is on the efficacy of particular equilibria at fixed discount rates. Barron andGuo (2020) also study communicative incentives with one-sided moral hazard, elucidatinghow public communication potentially exposes players to extortion.

Finally, this paper relates to our previous work, Ali and Miller (2016), and it is usefulto note the differences. Therein we study a society in which each pair of players plays asymmetric repeated Prisoner’s Dilemma. In that setting, every connection between playersis both a conduit for information and an opportunity for economic behavior. By contrast, ina networked market, an important distinction is between trading links (on which buyers and

3Another strand of the community enforcement literature studies folk theorems in anonymous settings.Deb and González-Díaz (2019) study how some cooperative outcomes are achievable in some games withoutcommunication, with a particular focus on games where there is one-sided moral hazard. Deb (2020) studiesgeneral games where players can announce names and authenticate them with their behavior, and Deb,Sugaya and Wolitzky (2020) prove a general folk theorem for anonymous random matching. Our workdiffers from this strand in that we study behavior with a fixed discount rate and focus explicitly on the roleof communication in a non-anonymous environment.

4Reputational labels and records also appear in work in theoretical biology that uses “image scoring” tounderstand indirect reciprocity; see, for example, Nowak and Sigmund (1998, 2005).

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sellers trade) and communication links (where traders on the same side of the market cantalk about their trading partners). While the negative result presented here is rooted in thesame strategic forces as that in our prior paper, our main contribution—the positive resultfor one-sided moral hazard—is fundamentally new. Hence, this contrast between one-sidedand two-sided moral hazard, which is our focus here, was absent in our previous work.

2 Model

2.1 The Setting

Society comprises buyers NB ≡ {1, . . . ,B} and sellers N S ≡ {1, . . . , S} where B,S ≥ 2. LetN ≡ NB∪N S. A generic buyer (“he”) is denoted by b; a generic seller (“she”) is denoted by s.The network of relationships in society features both trading links and communication links.Meetings between buyers and sellers occur on trading links: each buyer-seller pair bs meetsat random times in [0,∞) at Poisson rate λBS > 0. During these meetings, communicationand trade occur. Communication links involve meetings between players on the same side:each pair of buyers bb′ meets at Poisson rate λBB > 0, and each pair of sellers ss′ meets atPoisson rate λSS > 0, to communicate but not to trade. All meeting times are independentacross the network. Players share a common discount rate of r > 0.

This setting features “local monitoring”: if a pair of players is selected to meet at time t,only those two players observe the timing of their meeting and what transpires. Below wedescribe the extensive form in each of these meetings.

A buyer-seller interaction spans two stages: first the communication stage, in which theyexchange messages, and then the trading stage, in which they trade at a price p ≥ 0 anda quality q ∈ [0, q]. A buyer-buyer interaction or a seller-seller interaction has only thecommunication phase. We describe the trading stage first.

Trading Stage: At this stage, the buyer chooses a payment p ≥ 0 and the seller choosesthe quality of the good, q ∈ [0, q].5 When the buyer pays p and receives quality q, the buyer’spayoff is q − p and the seller’s payoff is p − c(q). We assume that c is strictly increasing andstrictly convex, that c(0) = c′(0) = 0, and that both q−c(q) and c(q)/q are strictly increasing.We consider three distinct extensive forms for the trading stage:

1. Simultaneous protocol: the buyer and seller make their choices simultaneously.

5The upper bound ensures that continuation payoffs remain bounded. We assume the bound is sufficientlyhigh that the constraint never binds.

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2. Buyer-first protocol: the buyer pays first, and upon receiving payment, the sellerchooses how much to deliver to the buyer.

3. Seller-first protocol: the seller delivers first, and upon receiving delivery, the buyerchooses how much to pay to the seller.

The simultaneous protocol exhibits two-sided moral hazard, since each partner has a myopicgain from deviating. The latter two protocols exhibits one-sided moral hazard, because theparty moving first can be immediately punished.

Communication Stage: When a pair interacts, along either a trading link or a commu-nication link, they engage in “polite cheap talk” where one of them is randomly selected withprobability 1

2 to speak first, and then after her speech, her partner speaks.6 The messagespace enables them to exchange information about which players have deviated. Specifi-cally, the message space is M ≡ 2N ; when player i sends message m in the equilibria we willconstruct, the interpretation is that player i is stating that m is the set of players who are“guilty” because they have deviated. Talk is cheap, so player i is free to send any messageregardless of his or her history.

2.2 Solution Concept: Permanent Ostracism

We study a class of equilibria that we call permanent ostracism equilibria. We describethe idea intuitively here, relegating formal details to the Appendix. A permanent ostracismstrategy profile is a pure strategy weak perfect Bayesian equilibrium, with an associatedsystem of beliefs, in which each player assesses others as being innocent or guilty. Thesereputational labels apply to those players that have a myopic gain from shirking, namely alltraders in the simultaneous protocol, sellers in the buyer-first protocol, and buyers in theseller-first protocol. In player i’s accounting, all such players begin as innocent. Player i mustdeem that partner j is innocent as long as player i has not obtained any indication to thecontrary. If player j has a myopic gain from deviating, then player i should reclassify player jas guilty if either of the following occur: (1) player j fails to exert expected effort or submitexpected payment when interacting with player i, or (2) any player k, when interacting withplayer i, sends a message m ∋ j. If player i deems partner j guilty, she permanently ceasestrading with partner j. Guilty or innocent, each player communicates truthfully about thebehavior of others, but not about herself. Off the equilibrium path, players’ beliefs reflect

6This sequential communication structure plays a role only in our negative result (Proposition 0) forthe reasons that we discuss in Ali and Miller (2016). Our main result (Proposition 1) obtains even ifcommunication is simultaneous.

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a correct understanding of the stochastic process governing how information about guiltdiffuses through the network given equilibrium behavior.

3 A Negative Result for Two-Sided Moral HazardWe first describe a negative result for trading games exhibiting two-sided moral hazard.The benchmark for this result is bilateral enforcement with a simultaneous protocol. Underbilateral enforcement, the behavior within each buyer-seller relationship depends only onpast interactions between them, independently of others. In this benchmark strategy profile,each time they meet, the buyer pays p and the seller chooses quality q; if either deviates, thepair responds by setting prices and quantities to zero in all future interactions. This grimtrigger punishment leads to the incentive constraints

q ≤ q − p + ∫∞

0e−rtλBS(q − p)dt, (Buyer’s Bilateral IC)

p ≤ p − c(q) + ∫∞

0e−rtλBS(p − c(q))dt. (Seller’s Bilateral IC)

Setting both inequalities to bind leads to the highest level of trade supportable by bilateralenforcement, q, which solves

c(q)q= ( λBS

r + λBS

)2

.

While multilateral enforcement could, in principle, improve upon bilateral enforcement, weshow that no permanent ostracism equilibrium can do better.

Proposition 0. With two-sided moral hazard, in every permanent ostracism equilibrium,the level of trade never exceeds q in any equilibrium path history.

Although Proposition 0 is not a formal corollary of our prior work (Ali and Miller 2016)—and hence, for completion, we include a proof in the Appendix—its logic is very similar: ifthe level of trade exceeds q, then Ann trusts Bob if she believes others are innocent andavailable to punish Bob. But if Bob knows that others have deviated, and divulging thisinformation to Alice will make her trust him less, he has no incentive to do so. He is betteroff concealing it and shirking on Ann. Since all he needs to do is conceal, the conclusionholds even if Bob has verifiable evidence that others have deviated.

We consider Proposition 0 as a benchmark. One may envision more elaborate schemesthat can do better than permanent ostracism equilibria in this setting. For example, tem-

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pering punishments with forgiveness may facilitate communication, as we emphasize in Aliand Miller (2016). Or alternatively, one may envision a scheme where bilateral enforcementis used to discipline one side of the market and permanent ostracism is used to discipline theother. Such modifications are fruitful with two-sided moral hazard, but as our result belowarticulates, unnecessary with one-sided moral hazard.

4 Positive Results for One-Sided Moral HazardThis section presents our main contribution: communication is incentive compatible at highlevels of trade if the stage game exhibits one-sided moral hazard. For concreteness, weconsider a buyer-first protocol; a similar result obtains for a seller-first protocol.

Before proving our result, we first describe a naive communication benchmark that wouldbe relevant if traders were forced to communicate truthfully about other traders. A playerwho is guilty will be punished by every partner he meets who deems him guilty. The best aguilty player can hope is to cheat every unsuspecting partner he meets. This logic leads tothe incentive constraints:

0 ≤ q − p + S ∫∞

0e−rtλBS(q − p)dt, (Buyer’s IC)

p + (B − 1)p vSB,S ≤ p − c(q) +B ∫∞

0e−rtλBS(p − c(q))dt. (Seller’s IC)

The Buyer’s IC reflects that so long as he is gaining from trade (q ≥ p), he must be betteroff from cooperating both myopically and in terms of discounted continuation value.

By contrast, the Seller’s IC reflects that the seller has a myopic incentive to deviatewhenever c(q) > 0, and her motive for not doing so is to maintain cooperation with futurebuyers. In Seller’s IC, vSB,S describes the (discounted) probability that when the seller nextmeets a given buyer, he has not yet learned of her guilt; naturally, vSB,S depends on the rateat which information about a guilty seller diffuses in the market.7 There are several waysfor information to diffuse: seller i may cheat on another unsuspecting buyer, or a buyeror seller who knows seller i is guilty may pass on that information.8 It is only in the firstcase that seller i accrues payoffs from others learning about her guilt, and vSB,S captures thediscounted probability of her obtaining a payoff from these future defections. Accordingly,the term (B − 1)pvSB,S measures the discounted value of seller i’s future opportunities to

7We show in Footnote 13 how to compute vSB,S recursively from primitives.8This benchmark uses both buyers and sellers as conduits of information about sellers’ behavior, so as

to maximize the level of cooperation. Later, we discuss equilibria in which only buyers communicate.

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cheat other buyers in the future, before they have learned that she is guilty.Setting these constraints to bind and simplifying yields the maximum level of cooperation

with naive communication under a buyer-first protocol, q∗BF, which solves

c(q)q=BλBS − (B − 1)rvSB,S

r +BλBS

. (1)

This is our benchmark for high cooperation. All of the “social collateral” here is put on theside of the sellers: because buyers have no myopic incentive to deviate, they can be deterredfrom deviating within the stage game and don’t need to be rewarded or punished throughcontinuation play. By contrast, the seller is deterred by multilateral enforcement, where sheputs several relationships at risk if she cheats in any one relationship. Thus, she is willingto produce higher quality in each relationship than under bilateral enforcement.9

But this benchmark is an unsatisfactory description of behavior because it assumes thatinnocent players are simply forced to reveal the truth. Our positive result is that this bench-mark is attainable, even if players can strategically choose what to disclose: we construct apermanent ostracism equilibrium that attains this naive communication benchmark q∗BF.

In this equilibrium, only sellers can be considered guilty; because a buyer has no incentiveto deviate, there is no reason to ostracize him. When a buyer meets a seller he believes isinnocent, he first pays p (regardless of whether he has deviated in the past). Then theseller produces quality q if she is innocent and price p was paid; she produces zero qualityotherwise. If the seller deviates in the trading stage, she becomes guilty, and thereafteraims to shirk on all future buyers too. Buyers, in turn, aim to ostracize guilty sellers whilecontinuing to trade at equilibrium-path levels with innocent sellers.

Players communicate truthfully, with the exception that each seller never communicatesabout herself. Consequently, an innocent seller’s incentive constraint is Seller’s IC fromabove, enabling cooperation at price and quality p = q = q∗BF both on and off the equilibriumpath. We show that a guilty seller has an incentive to cheat on every buyer once she hascheated on anyone.10 In this equilibrium, buyers have no incentive to deviate at the tradingstage. Moreover, neither sellers nor buyers have any incentive to lie.11 These observations

9This holds for both bilateral enforcement under a simultaneous protocol, which yields quality q, andfor bilateral enforcement under a seller-first protocol, which yields quality that solves c(q)

q= λBS

r+λBS.

10In the case of two buyers, this step is immediate. But once there are more than two buyers, an additionalstep is needed to show that a guilty seller does not wish to work so as to dampen the rate at which otherslearn that she has cheated in the past. Our argument adapts Ellison (1994) to show that if a seller isindifferent between working and shirking if she is innocent, she has a strict incentive to shirk if she is guilty.

11Thus the result holds regardless of whether communication is sequential or simultaneous.

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lead to our main result:

Proposition 1. With a buyer-first protocol, there exists a permanent ostracism equilibriumthat attains the naive communication benchmark.

The logic is that with a buyer-first protocol, an innocent seller meeting a buyer doesnot care whether that buyer has valuable relationships with other sellers—the buyer canbe trusted to pay in any case. Thus, we can make a buyer’s payoff from each interactionindependent of his message, which makes him willing to truthfully reveal whether othersellers have defected.12 Because buyers themselves are guarded—by having to move first—their communication guards the cooperation of others.

Prior to proving this result, we make a few remarks. First, consistent with the themeof personalized punishment, this equilibrium ensures that the presence of a few sellers whoalways cheat does not spoil collective cooperation. Second, it does not require coordinationon a common calendar or start date. Third, the same approach works for a seller-firstprotocol: a seller who deviates can be punished immediately by the buyer, while buyersare disciplined by permanent ostracism. A similar line of reasoning leads to an expressioncomparable to (1) for the highest level of supportable trade. Fourth, our model assumesthat sellers impose no externalities on each other. In many settings, sellers may generateexternalities, and hence cannot be trusted to report truthfully about each other. Buyerscan still be relied on, however, so one could construct an analogous equilibrium that usesonly buyer-to-buyer communication. Because information would spread more slowly, thesupportable level of trade would be lower than q∗BF. Fifth, the result would not extend toa model in which each player acts sometimes as a buyer and sometimes as a seller. In thatcase, even with a buyer-first protocol, a similar result to Proposition 0 would arise: a buyerwho knows that others have deviated can benefit from concealing what he knows in order toshirk when next he acts as a seller.

Proof of Proposition 1. We pair the strategy profile described above with a system of beliefsin which buyers believe that sellers are innocent until revealed to be guilty, either by theirown actions or via communication from other players. We consider the incentives of buyers,innocent sellers, and guilty sellers in turn.

Buyer Incentives: At every interaction, the equilibrium prescribes that a buyer commu-nicate truthfully; following communication, the buyer pays p = q∗BF to a seller he deems

12We note that while in our construction, the buyer obtains zero payoffs in each relationship, this is unnec-essary for communication incentives. All that is needed is that his payoff in any interaction is independentof his report, regardless of whether the seller with whom he is interacting is innocent or guilty.

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innocent, and pays zero to a seller he deems guilty. Because the buyer’s payment does notaffect continuation play (with either the same seller or any other), his incentive to pay per-tains only to the current period. When facing an innocent seller, he has no myopic incentiveto deviate, because any deviation leads the seller to subsequently deliver zero quality. Whenhe meets a seller he deems guilty, he expects the seller to deliver zero quality regardless ofhis payment, so there is no incentive to pay more than zero. Finally, the buyer does not gainby deviating in the communication stage of any meeting, because, regardless of his message,he obtains the same payoff in every interaction.

Innocent-Seller Incentives: Both on and off the equilibrium path, an innocent seller’s in-centive constraint to produce quality q = q∗BF is Seller’s IC, which is satisfied with equality: ineach case, she expects all buyers to continue communicating truthfully and cooperating withher regardless of the guilt or innocence of any other seller and regardless of the past devia-tions of any buyer. Similarly she expects all buyers and all other sellers, guilty or innocent,to communicate truthfully about her guilt or innocence. Finally, there is no contingencywhere a deviation from truthful communication improves her payoff.

Guilty-Seller Incentives: Equilibrium strategies prescribe that a guilty seller communi-cates truthfully about other sellers and produces zero quality. A guilty seller has no incentiveto misreport about other players, because her report doesn’t affect what happens in her cur-rent or subsequent interactions.

We now prove that a guilty seller finds it incentive compatible to produce zero quality.Suppose, without loss of generality, that seller s′ and buyer b′ meet at time 0, and seller s′

deviates and produces zero quality. Suppose that at time t, the seller meets buyer b′′. Ifseller s′ already knows that buyer b′′ deems her to be guilty—either because she has alreadydeviated in a prior meeting with b′′ or because b′′ told her so in the communication stage—then she has no incentive to deviate to higher quality. If instead b′′ deems her innocentand pays the equilibrium-path price p = q∗BF, then the seller could deviate once to choosingquality q∗BF (so as to delay buyer 1 from learning of her guilt). A sufficient condition for theseller to choose zero quality rather than deviate to q∗BF is if, for every kb ≥ 1 and ks ≥ 1,

q∗BF + q∗BFV (kb + 1, ks) ≥ q∗BF − c(q∗BF) + q∗BFV (kb, ks), (2)

where q∗BFV (kb, ks) is her expected continuation payoff when kb buyers and ks sellers (in-cluding herself) deem her guilty.13 Although seller s is uncertain about (kb, ks), if (2) holds

13The value of V (kb, ks) is computed from a recursive system of equations: for kb ∈ {1, . . . ,B} and

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pointwise for every realization of kb ∈ {1, . . . ,B} and ks ∈ {1, . . . , S}, then it is sequentiallyrational for her to produce zero quality. Observe that (2) can be re-arranged to

c(q∗BF)q∗BF

≥ V (kb, ks) − V (kb + 1, ks).

We verify that this inequality is satisfied for every kb ≥ 1 and ks ≥ 1. Because q∗BF bindsthe equilibrium path incentive constraint, a seller is just indifferent between the equilibriumpath and producing zero quality in every trading stage. Let q∗BFV (0,1) be her continuationpayoff if she has been on the equilibrium path until now but plans to shirk on the next buyershe meets. Then her binding Seller’s IC can be re-written as

q∗BF + q∗BFV (1,1) = q∗BF − c(q∗BF) + q∗BFV (0,1). (4)

Re-arranging (4) implies that

c(q∗BF)q∗BF

= V (0,1) − V (1,1).

Thus, (2) is satisfied if

V (kb, ks) − V (kb + 1, ks) ≤ V (0,1) − V (1,1). (5)

We prove that (5) is satisfied for all kb = 0, . . . ,B and ks = 1, . . . , S, adapting the argumentof Lemma 1 of Ellison (1994). We consider every sequence of link recognitions in which notwo links meet simultaneously, and then take expectations over them. Let ξ = (τz, ℓz)∞z=1 be asequence of link recognitions that take place in time span [0,∞), where (τz)∞z=1 is the orderedlist of link recognition times and (ℓz)∞z=1 is the list of links in their order of recognition. (Wedefine a link ℓ between player i and player j as a set {i, j}. But with some abuse of notation,we also say that ℓ ∈ A ×B if either (i, j) ∈ A ×B or (j, i) ∈ A ×B.)

ks ∈ {0, . . . , S}, let

V (kb, ks) = ∫∞

0

⎛⎜⎜⎝

e−rte−(λBBkb(B−kb)+λSS(ks−1)(S−ks)+λBSkb(S−ks)+λBSks(B−kb))t

⋅ (λBBkb(B − kb)V (kb + 1, ks) + λSS(ks − 1)(S − ks)V (kb, ks + 1)+ λBSkb(S − ks)V (kb, ks + 1) + λBSks(B − kb)V (kb + 1, ks) + λBS(B − kb)

)

⎞⎟⎟⎠dt, (3)

where V (B,ks) = 0 for each ks. Finally, let vSB,S ≡ V (1,1)/(B − 1). The recursive equation (3) documentshow the seller waits until the next interaction that diffuses information about her guilt, which happens onlywhen an “informed” trader meets an “uninformed” trader. The guilty seller reaps surplus only when shemeets an uniformed buyer, which occurs with density λBS(B − kb).

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Let K0 = (K0b ,K

0s ) be the initial “s-state”—the sets of buyers and sellers (respectively)

who deem seller s guilty, at a start time normalized to zero. Then, if the sequence of linkrealizations is ξ, the s-state immediately following the interaction at time τz is

(κzb(K0, ξ), κz

s(K0, ξ))

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(K0b ,K

0s ) if z = 0,

(κz−1b ∪ (ℓz ∩NB), κz−1

s ) if z > 0 and ℓz ∈ (NB ∖ κz−1b ) × (κz−1

b ∪ κz−1s ),

(κz−1b , κz−1

s ∪ (ℓz ∩N S)) if z > 0 and ℓz ∈ (N S ∖ κz−1s ) × ((κz−1

b ∪ κz−1s ) ∖ {s}),

(κz−1b , κz−1

s ) otherwise,

where, with some abuse of notation, we write (κz−1b , κz−1

s ) for (κz−1b (K0, ξ), κz−1

s (K0, ξ)).Define V (K0

b ,K0s ∣ ξ) to be the equilibrium continuation payoff of seller s when the initial

s-state is K0 = (K0b ,K

0s ) and the sequence of link recognitions is ξ. The change in seller s’s

continuation payoff when one more buyer j deems her guilty at the outset, for any j ∈ NB,

V (K0b ,K

0s ∣ ξ) − V (K0

b ∪ {j},K0s ∣ ξ)

=∞∑z=1

e−rτz ∑b∈NB

q∗BFI(ℓz = {s, b} and b ∈ κz−1b ((K0

b ∪ {j},K0s ), ξ) ∖ κz−1

b ((K0b ,K

0s ), ξ))

≤∞∑z=1

e−rτz ∑b∈NB

q∗BFI(ℓz = {s, b} and b ∈ κz−1b (({j},{s}), ξ) ∖ κz−1

b ((∅,{s}), ξ))

= V (∅,{s} ∣ ξ) − V ({j},{s} ∣ ξ),

(6)

where I is the indicator function. The weak inequality follows from

Kzb ((K0

b ∪ {j},K0s ), ξ) ∖Kz

b (K0, ξ) ⊆Kzb (({j},{s}), ξ) ∖Kz

b ((∅,{s}), ξ),

since, for fixed ξ, the set of players who learn about seller s’s deviation via a path throughbuyer j is decreasing in the number of other players who initially know of her deviation.

Observe that V (∣K0b ∣, ∣K0

s ∣) = EξV (K0b ,K

0s ∣ ξ). Therefore, since (6) holds for almost

every ξ, taking the expectation over ξ yields (5). □

5 DiscussionIn markets where traders cannot contractually commit to their terms of trade, word-of-mouth communication is viewed to be a powerful incentive: traders may stop trading with

13

those revealed to be defectors while continuing to trade with non-defectors. We begin withthe premise that traders may not truthfully communicate who is guilty unless they have anincentive to do so. Based on this premise, we find that markets in which traders on only oneside have a myopic incentive to shirk can support significantly higher volumes of trade thanthose in which traders on both sides face moral hazard. The rationale is that traders wholack a myopic incentive to shirk become “guardians” who communicate truthfully to others.Their truthful communication deters traders on the other side of the market from defecting.Below, we discuss implications of our analysis as well as important caveats.

Implications: While our model is stylized, these results may help us better understandwhen ostracism succeeds or fails in practice. Certain situations naturally take the form ofa sequential-move game. For example, in the long-distance trade model proposed by Greif(1993), merchants first decide whether to trust agents, and agents later decide to rewardor exploit that trust. Greif assumes that merchants share information with one anothermechanically, and permanently ostracize any shirking agent. Our results imply that even ifmerchants can choose to misreport, they would have no incentive to do so because of thesequential nature of the trading relationship.

In financial lending, a lender first decides how much to lend, and a borrower then decideswhether to repay. Hoffman, Postel-Vinay and Rosenthal (2019) document the expansion oflong-distance lending arrangements in 19th-century France, and show that it was supportedby the information circulated among notaries about creditworthiness of borrowers. Applyingour findings to their setting suggests that the success of these credit markets was facilitatedby the incentives for notaries, acting as agents for lenders, to communicate truthfully aboutthe deviations of borrowers.14

Bernstein (2015) documents a network of relationships among original equipment man-ufacturers (OEMs) and their suppliers. Within each OEM-supplier relationship, supplierbehavior is contractually specified in great detail but OEM behavior is not. Thus, a sup-plier has no legal recourse if the OEM steals its innovation and then puts production outfor bid. Recognizing this problem of one-sided moral hazard, one OEM formed a “SupplierCouncil” to promote communication among suppliers. Through the lens of our model, weinterpret this setting as a seller-first protocol (where OEMs are buyers), and the Council asa communication device that increases the rate of communication among sellers.

14Hoffman, Postel-Vinay and Rosenthal (2019) model borrowers as mechanically either having either goodor bad credit, and focus on the moral hazard problem of notaries over whether to refer borrowers with badcredit to other notaries. They take for granted that notaries will communicate truthfully about deviationsby their peers.

14

In other contexts, one may envision markets where enforcement intermediaries mitigateincentive issues on one side, and informal enforcement is used on the other side. For instance,in supply contracts where quality is not legally enforceable, buyers are often given the rightto withhold payment if they deem the quality delivered to be “non-conforming.” Similarly,franchising arrangements impose detailed, legally enforceable requirements on the detailsof franchisees’ business operations, but impose few requirements on franchisors (Blair andLafontaine 2011). Such arrangements enable multilateral enforcement because parties whono longer have a myopic incentive to deviate are truthful conduits of information. Thus,rather than substituting for community enforcement, we see a complementarity betweenformal and informal enforcement.15

Speaking to the difference between one-sided and two-sided moral hazard, Bolton, Greinerand Ockenfels (2013) discuss how before eBay payments were made through Paypal, bothbuyers and sellers acted simultaneously and either could deviate. To address this issue,eBay featured a two-sided feedback system but this failed to produce reliable reviews anddiscipline players. Once it was feasible to structure payments through Paypal, so that buyersno longer needed to be rated, a one-sided feedback system has remained, and such feedbackinfluences sellers’ payoffs.

Caveats: Our stylized model omits several considerations, and here, we discuss caveatsthat are important to consider in interpreting our analysis.

First, we abstract from other motives to conceal information. For the buyer-first protocol,we construct an equilibrium in which a buyer’s message does not influence his future inter-actions, which makes it incentive-compatible to communicate truthfully. In certain settings,communication may be exogenously costly, or a buyer may fear retribution from the seller.Addressing these issues would require a construction that offers either an offsetting benefitto buyers (such as a reward for communicating) or uses incentives similar to Chassang andMiquel (2019) so that a seller cannot perfectly attribute communications to a single buyer.A different issue, raised by Barron and Guo (2020), is that if buyers are being relied upon forcommunication in a buyer-first protocol, a buyer can potentially threaten to communicatethat the seller is guilty unless the seller provides a high quality product. They show thatthis prospect for extortion can not only destroy incentives to communicate truthfully butalso cause a breakdown in cooperation altogether.16

15Our point complements Acemoglu and Wolitzky (2020), who show how community enforcement cansubtly improve enforcement intermediation whereas we focus on the reverse channel.

16They study settings in which the party communicating can commit to certain communication strategies,which facilitates extortion. In their setting without commitment, there also exist equilibria in which truthful

15

Second, we focus on the class of permanent ostracism equilibria, and do not compare theentire set of equilibria between one-sided and two-sided moral hazard for a fixed discountrate. Our motive for this restriction is that permanent ostracism matches an intuitive notionof using communication for personalized punishment, which has been the focus of most priorwork on informal enforcement.17 But permanent ostracism has a number of defects. Onemay view it to take the principle of “ostracizing the guilty, cooperating with the innocent” toa logical extreme: perhaps after several individuals have been ostracized, it need not be thecase that innocent players continue to trade with other innocent players. Moreover, as weemphasize in Ali and Miller (2016), tempering punishments with forgiveness may facilitatecommunication, so punishments need not be permanent.18 Our results suggest that fromthe standpoint of communication incentives, combining permanent ostracism with thesemodifications would be fruitful in settings with two-sided moral hazard. But with one-sidedmoral hazard, such modifications are not needed to induce truthful communication.

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and Rasmusen (1989), Raub and Weesie (1990), Bendor and Mookherjee (1990), Klein (1992), Greif (1993),Ahn and Suominen (2001), and Dixit (2003, 2004).

18This is one motive for tempering permanent punishments in this context, complementing other rationalessuch as renegotiation-proofness and imperfections in monitoring.

16

Blair, Roger D. and Francine Lafontaine, The Economics of Franchising, Cambridge, UK:Cambridge University Press, 2011.

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Hoffman, Philip T., Gilles Postel-Vinay, and Jean-Laurent Rosenthal, Dark MatterCredit: The Development of Peer-to-Peer Lending and Banking in France number 92. In ‘ThePrinceton Economic History of the Western World.’, Princeton, N.J.: Princeton University Press,2019.

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17

Klein, Daniel B., “Promise keeping in the great society: A model of credit information sharing,”Economics & Politics, 1992, 4 (2), 117–136.

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Appendix A Definition of Permanent OstracismIn a permanent ostracism equilibrium, each player i has a personal state variable, ωi ⊂ N that liststhe players that i deems guilty. Player i’s behavior in each interaction depends on the history in away that is measurable with respect to ωi.19 For brevity, we define permanent ostracism only forsimultaneous and buyer-first protocols.

At the start of the game, ωi = ∅. Under a simultaneous protocol, any player can becomeguilty. However, under a buyer-first protocol, buyers cannot become guilty (they have “immunity”),because only sellers are subject to moral hazard. We write the set of players with immunity as I = ∅for a simultaneous protocol, and I = NB for a buyer-first protocol. When a buyer b and a seller smeet, and their personal states at the start of the trading stage are ωb and ωs respectively, thenunder a simultaneous protocol the buyer should pay p∗bs(ω

b) and the seller should deliver qualityq∗bs(ω

s). Under a buyer-first protocol, the buyer should pay p∗bs(ωb), and then seller should deliver

quality q∗bs(ωs) if the buyer paid correctly, but deliver quality zero otherwise. When player i meets

player j at time t, his personal state updates at the end of each stage of the interaction. We writeωi− for his state at the start of the stage, and ωi+ for his state at the end of the stage. At the endof the communication stage, after the partners exchange messages mi and mj , i’s state updatesfrom ωi− to ω+i = (ω−i ∪mi ∪mj)∖I. Then, again at the end of the trading stage, the state updatesfrom ωi− to ωi+ as follows, for each ℓ ∈ {i, j}:

• For a simultaneous protocol: If ℓ ∉ I and player ℓ plays any action other than p∗(ωi−) (ifplayer ℓ is the buyer) or q∗(ωi−) (if player ℓ is the seller), then ℓ ∈ ωi+;

• For a buyer-first protocol: If player ℓ is the seller and either (1) the buyer paid p∗(ωi−) andℓ delivers quality not equal to q∗(ωi−), or (2) the buyer paid any amount other than p∗(ωi−)and ℓ delivers quality not equal to zero, then ℓ ∈ ωi+;

• Otherwise ℓ ∈ ωi− ⇐⇒ ℓ ∈ ωi+.

Definition 1. An assessment (a strategy profile and a system of beliefs) is a permanent ostracismassessment if there exists a price function p∗bs ∶ 2

N → R+ and quality function q∗sb ∶ 2N → [0, q] for

each buyer-seller pair sb; each player i’s personal state ωi evolves according to the rule given above;and for every player i and every partner j ≠ i, if i meets j at time t, the following are satisfied:

1. In the communication stage, i sends the message ωi ∖ {i}.2. In the trading stage, if the protocol is simultaneous,

(a) if {i, j} ∩ωi = ∅ then i pays p∗ij(mi ∪mj) (if i is the buyer) or delivers q∗ij(mi ∪mj) (ifi is the seller);

19Although ωi is a function of player i’s private history, we suppress the history argument except whereneeded for clarity.

19

(b) if j ∈ ωi, then i pays 0 (if i is the buyer) or delivers 0 (if i is the seller).

3. In the trading stage, if the protocol is buyer-first:

(a) if i is the buyer: i pays p∗ij(mi ∪mj) if j ∉ ωi, but pays zero otherwise;(b) if i is the seller: i delivers q∗ij(mi ∪mj) if i ∉ ωi and j paid p∗ij(mi ∪mj), but delivers

zero otherwise;

4. When player i’s state is ωi at the start of any communication or trading stage when interactingwith player j, player i assigns probability 1 to the event that ωj ⊆ ωi.

The requirement on beliefs (Item 4) embodies ostracism: as long as player i has seen noindication—either directly or via messages from other players—that player k may have deviated,i should not believe that k has deviated and caused other players to deem k guilty.

B Proof of Proposition 0Proof. Suppose towards a contradiction that there is an interaction between players i and j atwhich their private histories at the start of the communication stage are (hi, hj), their personalstates are (ωi(hi), ωj(hj)), and they exchange messages mi = ωi(hi) and mj = ωj(hj), such that atthe start of the trading stage both i and j deem both i and j innocent, and q∗(ωi(hi)∪ωj(hj)) > q.

Consider another private history hi that coincides with hi except that every player other thani and j has transitioned to being deemed guilty by player i (so ωi(hi) = N ∖ {i, j}) after thelast interaction in hi. Suppose player j communicates first and sends message mj = ωj(hj). Ina permanent ostracism equilibrium, player i deems player j innocent, and so should report mi =ωi(hi) = N ∖ {i, j} truthfully. Then they should trade at quality q = q∗(N ∖ {i, j}) and pricep = p∗(N ∖ {i, j}). Note that q ≤ q and p ≤ p, since they must employ bilateral enforcement intheir relationship while permanently ostracizing all other players. However, if player i is the buyer,falsely reporting ωi(hi) and then reneging on payment yields a payoff of

q∗(ωi(hi) ∪ ωj(hj)) > q = q − p + ∫∞

0e−rtλ(q − p)dt,

where the first inequality is by our supposition, the equality is by definition of q and p, This is acontradiction to this strategy profile being an equilibrium. □

20